Notice that the proof uses only the covariance property for the parabolic subgroup of affine conformal diffeomorphisms of \(\mathbb {R}^{n-1}\) . The full covariance condition implies further conditions on the coefficients \(a_{i,j}\) , explicitly written by A. Juhl (see [1]}, condition (5.1.2) for \(N\)  even and (5.1.22) for \(N\)  odd), proving in particular that there exists (up to a constant) a unique covariant differential operator. Now let
\(E_N(\lambda )=\sum _{j=0}^{[\frac{N}{2}]} a_j (\lambda ,N)\left(\frac{\partial }{\partial \xi _n}\right)^{N-2j} \Delta ^{\prime j},\) 
